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Find the next term in the sequence.
$41,12,163,94,365,256,\,\_\_\_\_$
A) $647$
B) $547$
C) $497$
D) $257$

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Last updated date: 23rd Jul 2024
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Answer
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Hint: As we can see that there is a pattern of the numbers. Number pattern is a pattern or sequence in a series of numbers. This pattern generally establishes a common relationship between all numbers. In this question we have to find that pattern that binds them all and then apply on them and then we find the next number on the basis of it.

Complete step by step solution:
We have been given a pattern here $41,12,163,94,365,256,\,\_\_\_\_$
We have to analyse the relation between these numbers. Since these are not any perfect squares or there is any addition of numbers between them.
Let us take the alternate series which means there are two series combined in one. The first one is $41,163,365$.
When we add $41 + 122$ we get $163$ and then when we add $163 + 202 = 365$. We should note that the difference between their differences is $80$.
Their differences are $202 - 122 = 80$.
In the second series we have $12,94,256$. Here again we can see that the difference between their differences is $80$ i.e. it can be written as $12( + 82) = 94( + 162) = 256$.
The differences are $162 - 82 = 80$.
So from the above we can write that the next pattern will be $365( + 282) = 647$.
Hence the required next number is (A) $647$.

Note:
In this kind of question we should always try to first find the pattern between the numbers. There are several patterns as there can be differences between the numbers as we can see in Arithmetic progression, there can be a perfect cube of the numbers. For example the series of numbers $8,27,64,125...$. These are the perfect cubes i.e. ${2^3},{3^3},{4^3},{5^3}$, so we can say that the next number in the pattern is ${6^3} = 216$.